AG19 What is a Group Action?

Definition and Examples of Group Actions. We show that every group action gives a #homomorphism from the group to a group of #permutations #grouptheory #abstractalgebra #groups Subscribe ‪@Shahriari‬ for more undergraduate math videos. 00:00 Introduction 00:30 Definition of a Group (   • AG09 What is a Group? Definition & Examples  ) 01:22 Discussion: How do we study groups? 02:02 Intuitive idea of an action 02:37 Prototypical examples of an action 03:03 Definition of an action 05:08 Example: Natural action of S_n on {1, ..., n} (   • AG04 What is the Symmetric Group S_n? Cycl...  ) 07:09 Example: Action of S_4 on subsets of size 2 of {1, 2, 3, 4} 09:16 Example: D_8 acts on the corners of a square (   • AG02 Symmetries Of a Square; D_8 dihedral ...  ) 10:49 Example: D_8 acts on the diagonals of a square 12:14 Lemma: When a group acts, every group element gives a permutation of the set (   • AG03 Permutations and Bijections form a Gr...  ) 13:30 Proof of the Lemma 16:10 Example: GL(2, R) acts on R^2 (   • AG08 General & Special Linear Groups; Size...  ) 17:35 Theorem: When G acts on a set, we get a group homomorphism from G to the group of permutations of the set 18:42 Proof of Theorem 22:31 Kernel of an action 23:45 Example: Actions show that S_4 has a subgroup isomorphic to D_8 (   • AG05 S_4 has a subgroup isomorphic to D_8  ) 25:48 Example: Actions show that a cyclic group of order 2 is a homomorphic image of D_8 27:16 Previous Example gives a glimpse of quotient groups 28:44 Take Home Message A series of lectures for a full course on undergraduate abstract algebra based on my book: Shahriar Shahriari, Algebra in Action, A Course in Groups, Rings, and Fields, American Mathematical Society, 2017. https://bookstore.ams.org/amstext-27/ An annotated list of Abstract Algebra Videos: https://pomona.box.com/s/06zu90ewikpn... YouTube Playlist:    • Abstract Algebra, an intro via Actions   Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award.